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1. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor ; Zumbrun, Kevin ( January 2022 , Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$ (0, +\infty) $\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $\end{document} stable manifolds of such equations, showing that \begin{document}$ L^2_{loc} $\end{document} solutions that remain sufficiently small in \begin{document}$ L^\infty $\end{document} (i) decay exponentially, and (ii) are \begin{document}$ C^\infty $\end{document} for \begin{document}$ t>0 $\end{document}, hence lie eventually in the \begin{document}$ H^1 $\end{document} stable manifold constructed by Pogan and Zumbrun.
2. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan ; Zhang, Yanzhi ( January 2022 , Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$ (- \Delta)^\frac{{ \alpha}}{{2}} $\end{document} for \begin{document}$ \alpha \in (0, 2) $\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$ {\mathcal O}(h^2) $\end{document}, while \begin{document}$ {\mathcal O}(h^4) $\end{document} for quadratic basis functions with \begin{document}$ h $\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$ \alpha \in (0, 2) $\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document} for \begin{document}$ m \in {\mathbb N} $\end{document} and \begin{document}$ 0 < l < 1 $\end{document}, our method has an accuracy of \begin{document}$more »{\mathcal O}(h^{\min\{m+l, \, 2\}}) $\end{document} for constant and linear basis functions, while \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document} for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

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3. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

   https://doi.org/10.3934/cpaa.2021169  

   Jolly, Michael S. ; Kumar, Anuj ; Martinez, Vincent R. ( January 2021 , Communications on Pure & Applied Analysis)

   This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$ \theta $\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$ u $\end{document} is of lower singularity, i.e., \begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}, where \begin{document}$ p $\end{document} is a logarithmic smoothing operator and \begin{document}$ \beta \in [0, 1] $\end{document}. We complete this study by considering the more singular regime \begin{document}$ \beta\in(1, 2) $\end{document}. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of themore »original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.

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4. The mean-field limit of the Lieb-Liniger model

   https://doi.org/10.3934/dcds.2022006  

   Rosenzweig, Matthew ( January 2022 , Discrete and Continuous Dynamical Systems)

   We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finitemore »mass, but only for a very special class of \begin{document}$ N $\end{document}-body initial states.

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5. A measure model for the spread of viral infections with mutations

   https://doi.org/10.3934/nhm.2022015  

   Gong, Xiaoqian ; Piccoli, Benedetto ( January 2022 , Networks and Heterogeneous Media)

   Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$ S $\end{document} and removed \begin{document}$ R $\end{document} populations by ODEs and the infected \begin{document}$ I $\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$ S $\end{document} and \begin{document}$ R $\end{document} contains terms that are related to the measure \begin{document}$ I $\end{document}. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.